# A body of mass M at rest explodes in three pieces, two of which of mass M/4 each are thrown off in perpendicular directions with velocities of 3m/s and 4m/s respectively. The third will thrown off with a velocity of ?

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Initial mass = M

initial velocity = 0

After explosion,

m1 = M/4

v1 = 3m/s i (i means along x-direction)

m2 = M/4

v2 = 4m/s j (j means along y-direction)

m3 = M-(M/4 + M/4) = M/2

v3 = V

As there is no external force,

momentum before explosion = momentum after explosion

⇒M(0) = m1v1 + m2v2 + m3v3

⇒ 0 = M/4 (3)i + M/4 (4)j + M/2(V)

⇒ 0 = M [ (3/4)i + 1j + V/2 ]

⇒ 0 = (3/4)i + 1j + V/2

⇒ V/2 = -[(3/4)i + 1j]

⇒ V = 2×-[(3/4)i + 1j]

⇒ V = -[ (3/2)i +(2)j ]

⇒**V = -(3/2)i -2j**

|V| = √[(3/2)² + 2²] =** 2.5m/s**

initial velocity = 0

After explosion,

m1 = M/4

v1 = 3m/s i (i means along x-direction)

m2 = M/4

v2 = 4m/s j (j means along y-direction)

m3 = M-(M/4 + M/4) = M/2

v3 = V

As there is no external force,

momentum before explosion = momentum after explosion

⇒M(0) = m1v1 + m2v2 + m3v3

⇒ 0 = M/4 (3)i + M/4 (4)j + M/2(V)

⇒ 0 = M [ (3/4)i + 1j + V/2 ]

⇒ 0 = (3/4)i + 1j + V/2

⇒ V/2 = -[(3/4)i + 1j]

⇒ V = 2×-[(3/4)i + 1j]

⇒ V = -[ (3/2)i +(2)j ]

⇒

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Momentum is a vector.

Initial momentum of M is zero. So resultant of vector sum of all three momenta of 3 pieces is zero. Let us find the resultant of the momenta of two masses M/4.

Momenta are 3 M/4 and 4M/4 = M. The angle between them is 90⁰.

So θ = 36.87⁰.

So the momentum of the mass M/2 = 1.25 M.

**Its velocity = 1.25 M / (M/2) = 2.5 m/sec**

** Its direction is 126.87⁰** from direction of M/4 having velocity 3 m/s and

143.13⁰ from the direction of M/4 having velocity 4 m/s

Initial momentum of M is zero. So resultant of vector sum of all three momenta of 3 pieces is zero. Let us find the resultant of the momenta of two masses M/4.

Momenta are 3 M/4 and 4M/4 = M. The angle between them is 90⁰.

So θ = 36.87⁰.

So the momentum of the mass M/2 = 1.25 M.

143.13⁰ from the direction of M/4 having velocity 4 m/s